Asymptotic Behavior For Parabolic-Elliptic Keller-Segel Chemotaxis System With Logistic Source

This paper deals with a parabolic-elliptic chemotaxis system with nonlinear sensitivity and logistic source under homogeneous Neumann boundary conditions, in a smooth bounded domain ? (?)Rn (n? 1). The functions (?/(s), g(s), f(s)) ? C1+? ([0,?)) × C1+? ([0, ?)) × C0 ([0, ?)) ? C1 ((0, oo)) satisfy the following assumptions and respectively. We prove that for any M0> 0 if ||u0||L??M0, b large enough and q,k,l satisfy some addition conditions, then the solution (u, v) converges to the positive constant equilibrium (?).